Home page for accesible maths 4.1 Inference for a single proportion 4.1 Inference for a single proportion 4.1.2 Confidence intervals for a proportion

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4.1.1 Identifying when the sample proportion is nearly normal

A sample proportion can be described as a sample mean. If we represent each ‘‘success’’ as a 1 and each ‘‘failure’’ as a 0, then the sample proportion is the mean of these numerical outcomes:

\displaystyle\hat{p}=\frac{\ 0+1+1+\cdots+0\ }{976}=0.44

The distribution of $\hat{p}$ is nearly normal when the distribution of 0’s and 1’s is not too strongly skewed for the sample size. The most common guideline for sample size and skew when working with proportions is to ensure that we expect to observe a minimum number of successes and failures, typically at least 10 of each.

Conditions for the sampling distribution of $\hat{p}$ being nearly normal The sampling distribution for $\hat{p}$ , taken from a sample of size $n$ from a population with a true proportion $p$ , is nearly normal when 1. the sample observations are independent and 2. we expected to see at least 10 successes and 10 failures in our sample, i.e. $np\geq 10$ and $n(1-p)\geq 10$ . This is called the success-failure condition. If these conditions are met, then the sampling distribution of $\hat{p}$ is nearly normal with mean $p$ and standard error $\displaystyle SE_{\hat{p}}=\sqrt{\frac{\ p(1-p)\ }{n}}.$ (4.1)

Typically we do not know the true proportion, $p$ , so we substitute some value to check conditions and to estimate the standard error. For confidence intervals, usually $\hat{p}$ is used to check the success-failure condition and compute the standard error. For hypothesis tests, typically the null value – that is, the proportion claimed in the null hypothesis – is used in place of $p$ . Examples are presented for each of these cases in Sections 4.1.2 and 4.1.3.

TIP: Reminder on checking independence of observations If data come from a simple random sample and consist of less than 10% of the population, then the independence assumption is reasonable. Alternatively, if the data come from a random process, we must evaluate the independence condition more carefully.